Partial Differential Equations

The following sections are included:

• Preface

• Preface to the First Edition

• Contents

The following sections are included:

• Introduction

• Linear First-order Equations

• The Cauchy Problem for First-order Quasilinear Equations

• General Solutions of Quasi-linear Equations

• Fully-nonlinear First-order Equations

The following sections are included:

• Linear Equations

• Classification and Canonical Forms of Equations in Two Independent Variables

• Classification of Almost-linear Equations in Rⁿ

The following sections are included:

• The Wave Equation on the Whole Line. D'Alembert Formula

• The Wave Equation on the Half-line. Reflection Method

• Mixed Problem for the Wave Equation

• Inhomogeneous Wave Equation

• Conservation of the Energy

The following sections are included:

• Maximum-minimum Principle for the Diffusion Equation

• The Diffusion Equation on the Whole Line

• Diffusion on the Half-line

• Inhomogeneous Diffusion Equation on the Whole Line

The following sections are included:

• Weak Derivatives and Weak Solutions

• Conservation Laws

• Burgers' Equation

• Weak Solutions. Riemann Problem

• Discontinuous Solutions of Conservation Laws. Rankine–Hugoniot Condition

The following sections are included:

• Harmonic Functions. Maximum-minimum Principle

• Green's Identities

• Green's Functions

• Green's functions for a Half-space and Sphere
  • Half-space
  • Sphere

• Harnack's Inequalities and Theorems

The following sections are included:

• Fourier Series
  • Fourier coefficients. Convergence of Fourier series
  • Even and odd functions. The complex form of the full Fourier series

• Orthonormal Systems. General Fourier Series
  • The Bessel functions
  • Legendre polynomials

• Fourier Method for the Diffusion Equation
  • Homogeneous equation and boundary conditions
  • Inhomogeneous equation and boundary conditions

• Fourier Method for the Wave Equation
  • Homogeneous equation and boundary conditions
  • Inhomogeneous equation and boundary conditions

• Fourier Method for the Laplace Equation
  • BVPs for the Laplace equation in a rectangle
  • Dirichlet problem for Laplace equation in a disk

The following sections are included:

• Diffusion Equation in Three Dimensional Space

• Fourier Method for the Diffusion Equation in Higher Dimensions

• Kirchoff's Formula for the Wave Equation. Huygens' Principle
  • Kirchoff's formula. Spherical means
  • Wave equation on R². Method of descent. Huygens' principle

• Fourier Method for the Wave Equation on the Plane. Nodal Sets

The following sections are included:

• References

• Answers and Hints to Exercises

• Index